What are the benefits of the different cylinder caps?
I like the topology clean whether it's game model or not - so here are all the caps you will ever need:



If you have more sides, you can downscale these caps into them - the corresponding sub-cap topologies are highlighted (4-side cap, 6-side cap, 8-side cap):



With this you can make very clean any size cap with grid-like topology (which is very good).

If the number of sides is too big, the marked quads on next example will be a problem. This is how to setup loops in such topology to keep it again mostly grid and fix the problem:



Not only for subdivision topologies it's always good to inset the cap like so (its good for smooth shading also, thus for normal-maps etc.):





I don't like the square approach for triangles, I would prefer the vertex-to-vertex quad approach if I had. Long thin quads are bad also but still better than triangles.

Creating a triangle fan could be only pardoned in mobile game models or when flat shaded. Long thin triangles and a vertex with more than 5 edges are both the worst from topology and shading standpoint.